Saddle point braids of braided fibrations and pseudo-fibrations
Let gt be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt are distinct for all t, they form a braid B1 on n strands. Likewise, if the critical points of gt are distinct for all t, they form a braid B2 on n-1 strands. In this paper we study the r...
| Autores: | , |
|---|---|
| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Recursos: | Consejo Superior de Investigaciones Científicas (CSIC) |
| Repositorio: | DIGITAL.CSIC. Repositorio Institucional del CSIC |
| OAI Identifier: | oai:digital.csic.es:10261/381553 |
| Acesso em linha: | http://hdl.handle.net/10261/381553 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85190696117&doi=10.1007%2fs40687-024-00446-x&partnerID=40&md5=9c80691e3947731c99f4c5eadafef7f1 |
| Access Level: | acceso abierto |
| Palavra-chave: | 14J17 14P25 30C10 32S55 57K10 Braided open book Homogeneous braid Isolated singularity P-fibered braid Real algebraic link Saddle point braid |
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Saddle point braids of braided fibrations and pseudo-fibrationsBode, B.Hirasawa, M.14J1714P2530C1032S5557K10Braided open bookHomogeneous braidIsolated singularityP-fibered braidReal algebraic linkSaddle point braidLet gt be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt are distinct for all t, they form a braid B1 on n strands. Likewise, if the critical points of gt are distinct for all t, they form a braid B2 on n-1 strands. In this paper we study the relationship between B1 and B2. Composing the polynomials gt with the argument map defines a pseudo-fibration map on the complement of the closure of B1 in C×S1, whose critical points lie on B2. We prove that for B1 a T-homogeneous braid and B2 the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links L1 and L2 there is a mixed polynomial f:C2→C in complex variables u, v and the complex conjugate v¯ such that both f and the derivative fu have a weakly isolated singularity at the origin with L1 as the link of the singularity of f and L2 as a sublink of the link of the singularity of fu. © The Author(s) 2024.B.B. is supported by the European Union’s Horizon 2020 research and innovation program through the Marie Sklodowska-Curie grant agreement number 101023017. M.H. is supported by Japanese Society for the Promotion of Science JSPS KAKENHI Grant Number JP18K03296. We would like to thank Mark Dennis for fruitful discussions and the referees for helpful comments.Peer reviewedSpringer NatureConsejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72]202520252024info:eu-repo/semantics/articlehttp://purl.org/coar/resource_type/c_6501Publisher's versioninfo:eu-repo/semantics/publishedVersionapplication/pdfhttp://hdl.handle.net/10261/381553https://www.scopus.com/inward/record.uri?eid=2-s2.0-85190696117&doi=10.1007%2fs40687-024-00446-x&partnerID=40&md5=9c80691e3947731c99f4c5eadafef7f1reponame:DIGITAL.CSIC. Repositorio Institucional del CSICinstname:Consejo Superior de Investigaciones Científicas (CSIC)Ingléshttps://doi.org/10.1007/s40687-024-00446-xSíinfo:eu-repo/semantics/openAccessoai:digital.csic.es:10261/3815532026-05-22T06:33:51Z |
| dc.title.none.fl_str_mv |
Saddle point braids of braided fibrations and pseudo-fibrations |
| title |
Saddle point braids of braided fibrations and pseudo-fibrations |
| spellingShingle |
Saddle point braids of braided fibrations and pseudo-fibrations Bode, B. 14J17 14P25 30C10 32S55 57K10 Braided open book Homogeneous braid Isolated singularity P-fibered braid Real algebraic link Saddle point braid |
| title_short |
Saddle point braids of braided fibrations and pseudo-fibrations |
| title_full |
Saddle point braids of braided fibrations and pseudo-fibrations |
| title_fullStr |
Saddle point braids of braided fibrations and pseudo-fibrations |
| title_full_unstemmed |
Saddle point braids of braided fibrations and pseudo-fibrations |
| title_sort |
Saddle point braids of braided fibrations and pseudo-fibrations |
| dc.creator.none.fl_str_mv |
Bode, B. Hirasawa, M. |
| author |
Bode, B. |
| author_facet |
Bode, B. Hirasawa, M. |
| author_role |
author |
| author2 |
Hirasawa, M. |
| author2_role |
author |
| dc.contributor.none.fl_str_mv |
Consejo Superior de Investigaciones Científicas [https://ror.org/02gfc7t72] |
| dc.subject.none.fl_str_mv |
14J17 14P25 30C10 32S55 57K10 Braided open book Homogeneous braid Isolated singularity P-fibered braid Real algebraic link Saddle point braid |
| topic |
14J17 14P25 30C10 32S55 57K10 Braided open book Homogeneous braid Isolated singularity P-fibered braid Real algebraic link Saddle point braid |
| description |
Let gt be a loop in the space of monic complex polynomials in one variable of fixed degree n. If the roots of gt are distinct for all t, they form a braid B1 on n strands. Likewise, if the critical points of gt are distinct for all t, they form a braid B2 on n-1 strands. In this paper we study the relationship between B1 and B2. Composing the polynomials gt with the argument map defines a pseudo-fibration map on the complement of the closure of B1 in C×S1, whose critical points lie on B2. We prove that for B1 a T-homogeneous braid and B2 the trivial braid this map can be taken to be a fibration map. In the case of homogeneous braids we present a visualization of this fact. Our work implies that for every pair of links L1 and L2 there is a mixed polynomial f:C2→C in complex variables u, v and the complex conjugate v¯ such that both f and the derivative fu have a weakly isolated singularity at the origin with L1 as the link of the singularity of f and L2 as a sublink of the link of the singularity of fu. © The Author(s) 2024. |
| publishDate |
2024 |
| dc.date.none.fl_str_mv |
2024 2025 2025 |
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info:eu-repo/semantics/article http://purl.org/coar/resource_type/c_6501 Publisher's version info:eu-repo/semantics/publishedVersion |
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article |
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publishedVersion |
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http://hdl.handle.net/10261/381553 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85190696117&doi=10.1007%2fs40687-024-00446-x&partnerID=40&md5=9c80691e3947731c99f4c5eadafef7f1 |
| url |
http://hdl.handle.net/10261/381553 https://www.scopus.com/inward/record.uri?eid=2-s2.0-85190696117&doi=10.1007%2fs40687-024-00446-x&partnerID=40&md5=9c80691e3947731c99f4c5eadafef7f1 |
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Inglés |
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Inglés |
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https://doi.org/10.1007/s40687-024-00446-x Sí |
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info:eu-repo/semantics/openAccess |
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openAccess |
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application/pdf |
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Springer Nature |
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Springer Nature |
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reponame:DIGITAL.CSIC. Repositorio Institucional del CSIC instname:Consejo Superior de Investigaciones Científicas (CSIC) |
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Consejo Superior de Investigaciones Científicas (CSIC) |
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DIGITAL.CSIC. Repositorio Institucional del CSIC |
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